TRANSACTIONSOF THE

AMERICAN MATHEMATICALSOCIETY
Volume 316, Number 1, November 1989

HARMONIC MEASURE
AND RADIAL PROJECTION

DONALD E. MARSHALLAND CARL SUNDBERG

Abstract. Among all curves in the closed unit disk that meet every radius,
there is one, /q > whose harmonic measure at the origin is minimal. We give
an explicit description of yo and compute its harmonic measure. We also give
a quadratically convergent algorithm to compute the harmonic measure of one
side of a rectangle at its center.

1. INTRODUCTION

Let IV be a domain in the plane and let E be a Borel subset of the boundary
of IV, d IV. The harmonic measure of E at z E IV (relative to IV) is the
solution to the Dirichlet problem in IV with boundary values 1 on E and 0 on
dIV\E . More precisely, let Xe(<P)= 1 for <pE E, XeÍV) - 0 for <pe dW\E.
Then the harmonic measure at z is

w(z, E, IV) = sup < u(z) : u is subharmonic in W and

limsupw(z) < Xßiv) for tp E divi .

If F is a Borel subset of the closure of IV, the harmonic measure of F at z

will mean the harmonic measure of Fr\d(IV\F) with respect to the component
of W\F containing z . See Ahlfors [1] for an introduction to this subject.
Harmonic measure is extremely useful for estimating the growth of analytic
and harmonic functions, see Garnett [6]. An early example is the Carleman-
Milloux problem [4, 12, 13]: Suppose that / is analytic and |/(z)| < M in
the unit disk D. Suppose further that \f(z)\ < m on a curve y that connects
the origin to dD . How large can |/(z0)| be at a given point z0 ? By the two
constant theorem [1], |/(z0)| < mw Mx~w where w - w(zQ ,y,D\y). (Milloux
[12] attributes a version of this fact to Carleman.) What is needed, therefore, is
a lower bound for w which depends only on the fact that y connects 0 to dD .
A more general version of this problem was solved independently by Beurling
Received by the editors January 5, 1988.
1980 Mathematics Subject Classification (1985 Revision). Primary 30C85, 31A15; Secondary
30C30, 30C70.
Key words and phrases. Harmonic measure, extremal length.
Research supported in part by NSF Grant DMS 8601467.
©1989 American Mathematical Society
0002-9947/89 $1.00+ $.25 per page

8!
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 82 D. E. MARSHALL AND CARL SUNDBERG

[3] and R. Nevanlinna [14]. Kakutani [9] proved later that w(z,E, W) equals
the probability that a Brownian traveler starting from the point z first hits d W
in the set E. Thus the "further away" the set E is from z, the smaller its
harmonic measure. Beurling's result fits this intuitive notion. It says that if
zQ>0,then w(z0,y ,D\y) is minimal when y is the radius [-1,0].
We wish to consider a variant of this problem, namely: Suppose that / is
analytic and \f(z)\ < M in D. Suppose further that \f(z)\ < m on a curve
y that meets every radius. In other words, for each 8, 0 < 8 < 2n, there
iff
is a point re E y, 0 < r < 1 . How large can |/(0)| be? Again, by the
two constant theorem, |/(0)| < mwM ~w where w = w(0,y ,D\y). If y is a
closed curve, then by the maximum principle, |/(z)| < m, i.e., w = 1 . What
is needed, again, is a lower bound for w(0, y, D\y).

Theorem 1. Suppose y is a continuum in the closed unit disk D that meets

every radius. Then the harmonic measure at the origin for y in D\y is at least
c0 = .977126698498665669.... This lower bound is achieved only for rotations
and reflections of the curve y0 given by yQ = y U y where y is the lower
half of the unit circle, {z: \z\ = l,Im(z) < 0}, and y is the image of the
half-hyperbola x'/3 -y =1/4, y < 0, x > 0, under the linear fractional
transformation ( 1 - z)/( I + z), z = x + iy (see Figure 3). The constant cQ is
equal to the harmonic measure at the center of a 1 : 3 rectangle for the two long
sides.
A version of Hall's lemma [10] states that if E c D and if E* - {e' : re'6 E
E, some r > 0} is the radial projection of E on 3D, then w(0,E ,D\E) >
cw(0,E* ,D) - c\E*\/2n , where |£"*| is the Lebesgue measure of E*. Unlike
Beurling's theorem, c ^ 1 . Fuchs [2, p. 493] has asked what the optimal
constant c is. Our result shows how close c is to 1 in a special case.
In §2, we use extremal length to determine the optimal curve y0 . In §3,
we give a more explicit description of y0 and give a quadratically convergent
algorithm to compute c0 . In a future paper, with different techniques, we will
if)
treat the more general problem where the curve y meets radii {re : 0 < r < 1}
with 0 < 8 < a < 2n . In [5] it is proved that if a < n then the minimal
harmonic measure is a/2n . These authors pose the problem of determining the
maximal a for which this remains true. Our analysis will yield a computation
of this extremal value. In §4, we show that the constant cQ in Theorem 1 is not
the optimal constant c in the problem mentioned in the previous paragraph. In
the process of showing c < cQ, we give an alternate proof of a result of Hayman
[7].
We would like to mention that numerical computations were indispensable at
various stages of this project, though the proof of Theorem 1 does not depend
upon them. The extremal curve [Figure 3] was drawn using the conformai
mapping technique given in Marshall and Morrow [11]. We would like to thank
J. Morrow for his assistance. It was only after viewing this picture that we

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 HARMONIC MEASURE AND RADIAL PROJECTION 83

discovered the simple formula for it given in §3. It was because that mapping
technique is so well suited to map regions D\y to D, that L. Carleson passed
this problem on to us, for which we would like to thank him.

2. Proof of Theorem 1
In the course of the proof of Theorem 1, we will need the notion of extremal
length. If F is a family of locally rectifiable arcs in a region U and if p is a
nonnegative Borel measurable function on U (such a function will henceforth
be called a metric), we define the /^-length of </>E F to be

L(cf>,p)= [ p\dz\
J¡p
and the p-area of U to be

A(U,p)= [ p2dA
Ju
where dA is the Lebesgue measure on U. The extremal length of F in U is
defined to be
XL,(F)
u = supinf (^4}
p <peF[A(U,p) J •
See Ahlfors [1] for an introduction to extremal length. We will use only three
elementary facts about extremal length. The first is that it is conformally in-
variant, i.e., if / is a conformai map of U onto an open set U' and if F1 is
the image of F then XL,(F) - Xiri(F'). Indeed, if w = f(z), and if p is a
metric on U' then p(w)\dw\ is transformed to the metric p(f(z))\f(z)\\dz\
on U. The second needed fact is a beautiful criterion due to Beurling; see [ 1]
for the extremality of a metric.
Theorem 2 (Beurling). A metric p0 is extremal for F if F contains a subfamily
FQ with the following properties:

(1) / pQ\dz\ = inf L(4>,p) forall<pEF0.

Jip <!>&F

(2) For real-valuedh in U: if I h\dz\ > 0 for all cpE F0 then / hp0dA > 0.
Jip Ju
Moreover, in this case, the metric p0 is (a.e. dA) the unique extremal metric,
up to multiplication by a positive constant.
The major difficulty in extremal length problems is to discover the extremal
metric. Once such a metric is found, Beurling's criterion is usually used to prove
it is extremal. We suggest the reader use Beurling's criterion to prove that p = 1
is the extremal metric for curves that connect opposite sides of a rectangle.
Since the uniqueness portion of this theorem is not explicitly stated in [1],
we shall include the proof for completeness.
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 84 D. E. MARSHALLAND CARL SUNDBERG

Let p be a metric normalized by

inf L(<p,p)= inf L(tp,p0).

Then
/ p\dz\ > / p0\dz\ for all <pE F0.
J (p J ip
Let h = p - pQ. By (2)
/ p\dA< j p0pdA.
Ju Ju
By the Cauchy-Schwarz inequality

/ p\dA< [ p2dA.
Ju Ju
This proves that p0 is extremal. If fu p dA = fv p0 dA , then

/ {p0- pfdA = 2 (p] - p0p)dA<0.

Ju Ju
Hence p0 —p a.e. dA.
The third fact needed in the proof is given in the following elementary lemma.
Lemma 3. Suppose T is a one-to-one, conformai map (either analytic or anti-
analytic) from U onto U suchthat T oT(z) = z on U. Suppose further that
the curve family F satisfies T(F) = F. To compute the extremal length of the
family F, it suffices to consider metrics p with p oT\T'\ = p.
Proof. Let p = p o T\T'\ and let px=(p + p)/2 . Note that

I p dA = j p dA and / p\dz\ = I p\dz\.

Ju Ju Jt(ip) Jip
This implies that

/ p]dA= [ p2/2dA+ [ pp/2dA< [ p2 dA

Ju Ju J Ju
and
/ px\dz\> min / p\dz\, / p\dz\
Jip Jip J T(<p)

Thus inf L(tp , px)2/A(U,px) > inf L(tp,p) /A(U, p). Since To T(z) — z , we
conclude that pxoT\T'\ —px .
We will now prove Theorem 1. To obtain a lower bound for w(0,y ,D\y),
we may suppose, by an approximation, that y is a piecewise smooth Jordan arc.
We will later show there is an extremal curve y0 which is piecewise smooth.
Indeed, if / is the conformai map of the component of D\y containing 0
(which is necessarily simply connected) onto the disk D, then there is an arc
J c dD suchthat /~'(z) tends to y and z approaches the interior of J and
/~ (z) approaches dD as z approaches the interior of dD\J. Let Y = {z E
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 HARMONIC MEASURE AND RADIAL PROJECTION 85

D: w(z,J,D) = 1 - e}, and let yx = f~ (Y). Clearly yx is smooth, meets

each radius and w(0,yx ,D\yx) - w(0,y,D\y)/(l - e). By altering Y slightly
near the ends of J , and rotating D, we may suppose that y is a smooth map
from the interval [0,1] into D with y(0) > 0. By taking a subarc of y, if
necessary, we may suppose that y intersects the interval (0,1] in exactly two
points: y(0) and y (I). By reparameterizing y, we may suppose y(0) > y(l).
There is another reduction, which comes from [5]. The proof can be com-
pleted without it, however it makes the proof a little easier. Let r = min{a E
[-1,0): a = y(t) for some /}, let E = [-1 ,r]u[y(0),l], and let IV = D\E .
Finally, let k be the conformai map of D\E onto D with k(0) —0 and k
real-valued on (-1,1). Clearly,
w(0,k(y),D\k(y)) = w(0,y,D/(yuE))<w(0,y,D\y).
Note that k(y) meets every radius, and meets dD in at least two points. If
y(t0) E dD and y(tx) E dD, /0 < /, , then we may replace y on the interval
[Z0,/[] by an arc on dD from y(tQ) to y(tx), resulting in smaller harmonic
measure, by the maximum principle. The resulting curve will still meet every
radius since y begins and ends on the interval (0,1]. Replacing y with y =
{z: z e y}, if necessary, we may suppose that Im(y(/)) < 0 for / < e and
Im(y(t)) > 0 for / > 1 - e. In other words, we may suppose that the curve
y begins at 1, follows the unit circle clockwise at least as far as the point -1 ,
ends at y (I) > 0, meets each radius, and crosses the interval (-1,1) between
-1 and 0 (if at all). (See Figure 1.)

Figure 1

Let / be the conformai map of D\y onto D, with f(0) = 0, where

f(dD\y) consists of an arc a on dD centered at 1. By the conformai in-
variance of harmonic measure, and the Poisson integral formula,
w(0,y,D\y)= 1 - \a\/2n
where \a\ denotes the length of the arc a . So to minimize w(0,y ,D\y), we
must maximize \a\ . Let F denote the family of rectifiable arcs cfi, defined for
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 86 D. E. MARSHALLAND CARL SUNDBERG

O < / < 1 , with 0(0) G a, 0(1) G a, cp(t) g D\{0} for 0 < / < 1 and such
that 0 and -1 are not in the same component of D\(cf>u a). In other words,
each cp encloses 0 and begins and ends on a . Clearly if we increase a , we will
increase the family F and hence decrease the corresponding extremal length.
Let C7„ denote the family {f~x(<p): cpe F}. These are curves that separate 0
from y and begin and end on dD\y . By the conformai invariance of X, our
problem is to find the curve that minimizes XDV(G).

Figure 2

Let Td be the trapezoid with vertices vx =0,v2 = d > 0, v3 = d+l-i, and

v4 = -i. There is a unique number d > 0 and corresponding conformai map
hd with maps Td onto the upper half disk D+ = {z: \z\ < 1 and Im(z) > 0}
with h(vx) = 0, h(v2) = -1, h(v3) = +1 and h(v4) = y(l). Let Sx be
the union of Td together with its reflection about the line Im(z) = 0. Let
S2 be the reflection of Sx about the line Im(z) = 1, let S3 be the reflection
of Sx about the line Re(z) = 0 and let S4 be the reflection of S3 about the
line Im(z) = -1 . Finally, let S = Sx U S2 U S3 U 54 (see Figure 2). The
region 5 is the union of 8 trapezoids congruent to Td . The map hd extends
analytically to 5 by the Schwarz reflection principle. Let UQ be the component
of int(5)\{z G S: h(z) e y} containing the origin, where int(5) is the interior
of S. Since y crosses (-1,1) only between -1 and 0, the boundary of U0
consists of the two curves {z g S: h(z) g y} together with two line segments
L, and L2 with L, G dS4 and L2 E dS2 , each with slope -1 . On the region
Z70, the map h is a two-to-one cover of D\y . To see this, note that h is two-
to-one on Sx U Sj, and that if z G S and z ±2i ES, then h(z) = h(z ± 2/).
Moreover, if z G U0 , exactly one of the points z, z + 2/, z - 2/ is in 5, U S3 ,
call it u(z). The function u extends to a one-to-one map of U0 onto Sx US3 .
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 HARMONIC MEASURE AND RADIAL PROJECTION 87

Thus h is two-to-one; in fact, h(—z) = h(z). Thus for each cf>E F, there
is a rectifiable arc y/ in U0 with endpoints on Lx U L2 that either "encloses"
the origin or else connects Lx and L2. Let Hd be the family in U0 of all
such curves. To compute the length of the family Hd, by Lemma 3, we need
only consider metrics p on U0 with p(-z) = p(z). Any such metric can
be written in the form p(z) = q(hd(z))\h'd(z)\ where q is a metric on D\y.
Clearly A(U0,p) = 2A(U0,q) and L(y/,p) = L(hd(\p),q), so that XUo(Hd)=
XD.(G)/2 . Thus we wish to minimize XUo(Hd).
Let p = 1 on i/0 . The p-area of Trf is d+l/2, so by use of the above map
u , JLI pdA —4(d + 1/2). Note that the distance from L, to the line through
the origin with slope -1 equals y/2 + d/y/2 . Thus every curve \¡i E Hd has
length

\dz\>2(y2 + ^j.
/J tp

We conclude
[2(y/2+ d/V2)]2
hS"d)> 4(^+1/2) -3-
The latter inequality is strict, unless d = 1 . Now let d — 1 and let o be
the straight line from v2 to vA. Let t be the bottom half of the unit circle
= {z: \z\ = 1 and Im(z) < 0}. Let y0 = t U h(o) (see Figure 3). The
corresponding U0 is then clearly a 3 by 1 rectangle whose sides have slope
4-1 and -1 (see Figure 4). Let HQ consist of the line segments of slope 1
connecting opposite sides of UQ. By Beurling's criterion, p0 = 1 on U0 is the
unique extremal metric for this family and hence Xv (H0) = 3. This proves
that rninAflv (67 ) = 6 and that y0 is extremal. Moreover, any other extremal
curve would necessarily have d = 1 . The proof also shows that the Euclidean
metric on UQ must be extremal. By Beurling's criterion, the extremal metric
always comes from the conformai map of U0 onto a rectangle. Hence U0 must
be a rectangle in the extremal case.
So far, we have supposed that the candidates for the extremal curve are
piecewise smooth. We would now like to prove that y0 is the only continuum
in D, meeting every radius, for which the harmonic measure is minimal. Let
y be such a continuum. Then y may be approximated, as indicated above, by
curves yn, n = 1,2, ... , with corresponding regions U0(yn) and conformai
maps hn of U0(yn) onto D\yn. Let kn be the conformai map of U0(yn) onto
a rectangle Rn so that pn = \k'n\ is the extremal metric for Xy , AHd ). Since
XUo,7n)(HdJ= 3 , we may choose a normalization for kn so that Rn converges
to the 3 by 1 rectangle U0(y0). Thus there is a subsequence {k~x} of {kn}
converging uniformly on compact subsets of U0(y0) to the map k(z) = z.
We conclude that the corresponding maps hn must converge uniformly on
compact subsets of t/0 to the map h0 of UQ onto D\y0. Thus y0 is the
unique extremal continuum. We note that the curve in Figure 3 beginning at
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 88 D. E. MARSHALL AND CARL SUNDBERG

Figure 3

Figure 4

the origin and extending to dD is, in some sense, the most efficient path from
the origin to dD\y . It is the image under the map h of the straight line in UQ
through 0 and with slope 1.
We remark here that our discovery of the extremal metric was motivated
by the deep work of Jenkins [8]. It is not hard to show that Jenkins' theorem
implies that the extremal curve (suitably normalized) must consist of an arc
from 1 to a point e on \z\ = 1 then an arc in D from e' to a point x E D,
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 HARMONIC MEASUREAND RADIAL PROJECTION 89

x > 0. Moreover, the extremal metric is given by

2 (z-e )
P =c:
z(z - l)(z - x)(z - l/x)
where c is a constant. Jenkins show that for each x there is a unique 8 so
that this metric corresponds to the curve, of minimal harmonic measure, that
ends at x and meets every radius. He does not have a formula for 8 in terms
of x. The difficulty is in deciding which x corresponds to the extreme case
in our problem. It may be instructive to note that the trapezoids Td that we
constructed do not correspond to these metrics, unless d = 1 . Indeed, one can
show that e'e ^ -1 if x is not the endpoint of our external y0 , yet the metrics
associated with the rectangles formed from Td will have a singularity at -1.
We do not know of any "nice" conformai maps corresponding to the metrics of
Jenkins. We will reexamine Jenkins' metrics in another paper, as mentioned in
the introduction.
3. The extremal curve
Let y0 be the extremal curve in §2, let h be the associated map of the
rectangle U0 onto D\y0, and let / be the conformai map of D\y0 onto D
with f(0) = 0, and with f(dD) equal to an arc a on dD centered at 1. If cpx
is a conformai map of D onto UQ with f o h o q>x(0)= 0 then cj>= f o ho <px
is a two-to-one map of D onto D with (f)(0) = </>'(0)= 0. An elementary
argument shows that <p(z) = cz2 where c is a constant. Hence y/f o h is a
conformai map of the rectangle U0 onto D, which maps the four vertices to
points ß,ß, - ß,ß . Note that the endpoints of the arc a are ß and ß ,
id
and hence w(0,y0,D\y0) = 1 - 2\8\/n where ß = e . Note that U0 has side
ratio 3:1. The quantity \8\/n is the harmonic measure, at the center, of one
of the short sides of a 3:1 rectangle. By the Schwarz-Christoffel formula, the
number 8 is the solution to the integral equation (0 < 9 < n/4) :
1/2 fJt/2 1 1 1/2
f i y1 . _ j ds.
Jo [cos2s - cos2öJ Je cos 28 - cos 2s
The solution of this equation by standard methods of numerical integration,
while possible, is not trivial and converges slowly. Trefethen [15] gives a rapidly
converging algorithm to compute 8 based on a different method. An algorithm
to compute 8 seems to be of some numerical interest. For example, Trefethen
[ 16] has shown the use of 8 in the design of a resistor. This is equivalent to
finding the parameter k, 0 < k < 1 , for which the Jacobi elliptic function
sn(z,k) maps a 1 by 3 rectangle onto the upper half-plane. We give here a
method that is very simple and converges even faster (quadratic convergence)
than Trefethen's method. It is based on the following observation. Suppose
RL is a rectangle centered at the origin with sides parallel to the axes, with
height 1 and length L > 1 . Let SL be the semi-infinite strip {z: Re(z) >
-L/2,|Im(z)| < 1/2}. Let wR (z) be the harmonic measure of the left-hand
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 90 D. E. MARSHALLAND CARL SUNDBERG

edge of RL , {z E RL: Re(z) = -L/2}, at the point z and let ws (z) be the
harmonic measure of the left edge of SL, {z E SL: Re(z) = -L/2}, at the
point z. Clearly, wR (z) < ws (z) by the maximum principle. On Re(z) =
L/2 , wSl(z) < wSl(L/2) . Thus''for z G dRL
wSl(z)+wSl(-z)<{1+wSl(L/2)}{wRl(z) + wRl(-z)}.
By the maximum principle, this inequality holds at z —0. We conclude that

-Sl
wsJ0)
/T ... < wR (0) < w^ (0).
(1 1 -r-UIç (L/2) RLy Sl '

Note that wSl(L/2) = ws (0). By explicitly mapping SL onto the upper-half

plane and using elementary trigonometric identities, it is easy to show that
-1, -jcL/2.
ws(0) = ±tan \e
Hence
-1/
tan (e -TtL/2, ) -nL/2
(2) <wRL(0)<±tan \e
1 + 4 tan ■X(e-«L)

Note that (2) says wR (0) ^e n as L tends to oo. We can improve

this estimate in the following way. The quantities «^ (0) and wR2L(0) are
related as follows: Let q>Lbe the conformai map of RL onto D so that the
:fl _ ;Q :n _¡ft
images of the four vertices are e L, e L, - e L, - e L, for some number
8L, 0 < 9L < n/2, and <p'L(0)> 0. Note that <pL is real on R n RL and
tpL(0) = 0. Moreover, wR (0) = 8L/n. Note that <pL maps the top half of
RL onto the upper half disk D+ . Let x be the conformai map of D+ onto
D so that x{eif>L)= J(l) = -^(-1) = -x(-e~i&L). The top half of RL is
clearly the image under the map z » (z + i'/2)/2 of R2L. Hence (p2L(z) =
if)
X(<PL((z + i/2)/2)) and hence e' 2L X(e L). An explicit computation of the
map x~l shows that
'-A 171WR (0)
sin 7iu;
Sl(0) = 1 + sinnwR (0) '
Hence, if we let
-1/2
.v„ (sin7ruV,L(°))'
we have
V77-l
ÎK+!MJ.
By(i;
w (0)
0< < 2"+' L

^„,(0) wSl*L (0) wSl»L (0)

By elementary calculus, if
■1/2
(sinnws (0))'
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 HARMONIC MEASURE AND RADIAL PROJECTION 91

then 0 < xn - y < c(e~K ) , for some constant c. Note that xn > xn_x > 1
for all n , and that if y > 1,

\x„-x-\lh(y + y-)\<c\xn-y\
for some constant c (c = \ will suffice). Thus if we let yQ = y and yk —
J\{yk_x + y*—), 1 < k < n, then by induction

(3) \(sinnwRL(0))-x/2-yn\<C(e-nL/2f

where C is a constant which can be determined by elementary calculus. In

other words, the number of correct digits in the decimal expansion at least
doubles when n is increased by 1. To simplify the algorithm further, note that

(i+±AAA\
•>■->'-
and hence
-7i2"L
1 7t2"-2L e
yo~îe < —^—

Thus we may replace yQ with je71 and obtain the estimate (3) above with
slightly larger constant C. Finally

\wRL(0)-±sin-x(y-n2)\<C(e-nLI2f

for a slightly larger constant C. Notice that

sin_1(v"2) = tan"'
\yn-i ~l/yn-
Thus we obtain the following quadratically convergent algorithm:
Given n and L, let
i TtZ.2"-2
y0 = ze

yk = y/ïiy,c-i + l/yk-\)> i<k<n-i,

and
w = i tan
yn-i - [/yn-

Then
|w-(0)-«;|<C(e "L/2)2"
For example, with L = 3 and n = 2 the computed value for cQ = 1 - 2w =
.977126698498665669... is correct to 18 decimal places. This is virtually a
formula for cQ. We carried out the above estimate more precisely to obtain
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 92 D. E. MARSHALLAND CARL SUNDBERG

an error less than 10~'7 for any L > 1, when n — 4. Of course, for large L
one could (and should) reduce the number of steps. The astute reader can use
the above technique to give a quadratically convergent algorithm to compute
the conformai map of D to any rectangle if he so desires.
The optimal curve y0 (see Figure 3) was drawn using the conformai mapping
technique given in Marhsall and Morrow [11]. Although the map from the
trapezoid Tx to the upper half disk cannot be written in terms of elementary
conformai maps, the picture of y0 led us to believe that there might be a simple
formula for y0. Recall that Tx is the trapezoid with vertices 0,1,2-/ and
-/. Let Sx be the square with vertices 1,2 - i, 1 - 2/ and —i. Let fx be
the conformai map of Sx onto D such that fx(2 - i) — 1 ,fx(l) = /,/,(-/) =
-1 ,/,(l - 2Ï) = -i. Then /, maps Sx n 77, onto the upper half disk D+
and the right triangle T2 with vertices -/, 1 and 1 - / onto the left quarter
circle Q — {z: \z\ < l,Re(z) < 0,lm(z) > 0}. Reflecting T2 about the line
Re(z) = Im(z) and Q about the quarter circle on its boundary, we obtain a
conformai map of 77, onto the union of D+ and {z g C: Re(z) < 0,lm(z) >
0}. Applying the conformai map f2 = -l/z to this region, we obtain a
conformai map to C\(D+ u (-ex, - 1]. Apply the conformai map /3(z) =
(l + z)/(l-z) we obtain the region C\({z: Re(z) > 0 and Im(z) > 0}U[-1,0].
Applying the map f4(z) = (—iz) we obtain the region {z: Im(z) > 0}\/_3
where L3 is a straight line segment from the origin to the point e'n . The
function gx(z) = 3(z - l/3)1/3(z + 2/3)2/3/22/3 maps the upper half plane
conformally onto this latter region (see Marshall and Morrow [11] for details).
Note that g,(-l) = -1 . Let

£2(z) = (l-4z2)/3 and g3(z) = (I - z)/(l + z).

The upper half-plane is the image of the quarter plane Q = {z: Re(z) >
0,lm(z) < 0} under the map g2 and Q = g3(D+). Thus h = g3 o g~ o
g~l of4of3 of2ofx is the conformai map of 77, onto D+ suchthat h(0) = 0,
h(l) = -1 and h(2-i) - 1 . A direct computation shows that h(-i) = l-4yfi>.
The curve y0 n D is the image of the straight line segment L4 from -/ to 1
in Tx , under the map h . To find this curve more explicitly, first note that the
image of L4 under the map f4of3of2ofx is the ray {re' : 0 < r < oo} . To
find the image of this curve under the map g2 o g~ , we must find all z E Q
such that {gx °g2(z)Y is positive. In other words, 4z (1 - |z )" < 0. Taking
a square root, we want Re(4z(z -3/4)) = 0. Let z - cos(u) = (e'" + e~'")/2.
Then 4z(z2 - 3/4) = cos(3w) = cos 3x cosh 3y + i sin 3x sinh 3y . Thus we seek
those u for which Re(é> '") = 0. Hence <?'" = re' where 8 = ±n/6, ± n/2,
±5n/6, r > 0. Since z = cosd(r + l/r)/2 + isind(r - l/r)/2 E Q, we may
write z = y/3(r + 1/r)/4 + i(r - 1/r)/4 where 0 < r < 1 . If we write z —x + iy
then 4x2/3 - 4y =1 and y < 0. Finally we compute the image of this
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 HARMONIC MEASURE AND RADIAL PROJECTION 93

half-hyperbola under the map g3 = (1 - z)/(l + z). In terms of r it is

4r - V3(r2 + I) - i(r2 - I)
4r + y/3(r2+ l) + i(r2-l)'
Thus the optimal curve y0 begins at 1, follows the unit circle clockwise to
the point -1, then enters the unit disk D, making an angle of n/6 with the
positive x-axis, continues in the upper half disk D+ , until it meets R at right
angles at the point 7 - 4\/3 = .0718 .
4. On Fuchs' problem
In this section, we show that if we remove the hypothesis that y is connected,
the constant c0 is no longer the lower bound for the harmonic measure at the
origin. The proof below is motivated by the example in [7].
Suppose £ isa Borel subset of the closure of the right half-plane, R, and
suppose E contains a free boundary arc on the imaginary axis. In other words,
there exists a open disk D(ia, r) with center at ia , a real, and radius r > 0 so
that D(ia,r)nE = {it: a-r < t < a + r}. If 0 < Ô < r, let /, denote the line
segment from i(a + S) to ô + ia, let l2 denote a line segment from i(a + ô/2)
to i(a + a), and let E6 = E u /,\/2. In the case when E is the imaginary
axis, /, and a - 0, we denote this set by Is . The idea of the argument below
is that w(z,I,R) = 1 and for z fixed w(ôz,IS,R) = c < I, where c is
independent of ô . So for an arbitrary set E, at a point in R near ia, with ô
extremely small, E "looks like" / + ia and E& "looks like" Ig + ia , and thus
w(z ,Es,R\Eg) should be less than w(z ,E ,R\E).
Proposition. For each z E R\E, there is a SQ> 0 so that w(z,Eg,R\Es) <
w(z,E,R\E) when 3<ö0.
Proof. Without loss of generality, a = 0. By the maximum principle, it suffices
to show the inequality on fin {z: \z\ = 38} . Let o£ = {e'e: \e2'8 + 1| <
£, - ti/2 < 8 < n/2} and let oQ = {e'e : \eiB - 1| < 1/2}. Recall that /, is
the set formed from the imaginary axis with a = 0 and ô = 1 , by the process
described above. Let
e. = min(l - w(2e' ,I.,R\IA).
e,s€ao

By using an explicit conformai map of R\D(0,2) onto R, it is not hard to

show that
(4) w(z,2ae,R\D(0,2))< \exw(z,2o0,R\D(0,2))
when \z\ = 3, and e is sufficiently small.
Now for z = 28e with 8 fixed, w(z,E,R\E) —<■1 as ô —► 0 and
iß
w(z,Eg,R\Es) —*w(2e ,IX,R\IX) < 1. Thus we may choose ô > 0 so
that when z = 2ôe' with e' E oQ,
w(z,E,R\E)-w(z,Eg,R\Eg) > ¿e.
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 94 D. E. MARSHALLAND CARL SUNDBERG

and so that when z = 2ôe with e E dD(0,l)\ae

w(z,E,R\E) - w(z,Eg,R\Es) > 0.
Thus for \z\ = 23
w(z,E,R\E)-w(z,Eg,R\Eg)
>-w(z, 2âoe, R\D(0,26)) + \sxw(z, 2ôo0 , R\D(0,23)).
We conclude that for \z\ = 3ô , by (4) and the above,
w(z,E,R\E)-w(z,Eg,R\Eg) > (±e, - ±ex)w(z,2ôa0,R\D(0,2â)) > 0.
This proves the proposition.
An easy consequence of this proposition is the following result of Hayman
[7]. If E is a Borel subset of the right half-plane R, let E* = {iy: y > 0 and
if)
e y E E for some 8} .
Corollary 1. There is a set F with F* = {iy: y > 0} and w(l ,F* ,R) — j >
w(l,F,R).
To see this, apply the proposition with F — {Is + i}\{iy: y < 0} for ó
sufficiently small. We remark that this example doesn't really depend on the
exact nature of the set Is near 0.
If we let E denote the image in R of the extremal curve given in the previous
section, under the linear fractional transformation (1 + z)/(l - z), and then
apply the proposition, we obtain:
Corollary 2. // we do not assume that the set y in the statement of Theorem 1
is connected, then the harmonic measure of y at the origin can be smaller than
c0.

REFERENCES

1. Lars V. Ahlfors, Conformai invariants, McGraw-Hill, New York, 1973.

2. K. F. Barth, D. A. Brannan and W. K. Hayman, Research problems in complex analysis. Bull.
London Math. Soc. 16 (1984), 490-517.
3. A. Beurling, Études sur un problème de majoration, Thèse, Uppsala, 1933.
4. T. Carleman, Sur les fonctions inverses des fondions entières, Ark. Mat. Astronom. Fys. 15
(1921).
5. C. FitzGerald, B. Rodin and S. Warschawski, Estimates of the harmonie measure of a contin-
uum in the unit disk, Trans. Amer. Math. Soc. 287 (1985), 681-685.
6. J. B. Garnett, Applications of harmonic measure, Univ. of Arkansas Lecture Notes, vol. 8,
Wiley-Interscience, New York, 1968.
7. W. Hayman, On a theorem of Tord Hall, Duke Math. J. 41 (1974), 25-26.
8. J. Jenkins, On a problem concerning harmonic measure, Math. Z. 135 ( 1974), 279-283.
9. Shizuo Kakutani, Two dimensional Brownian motion and harmonic functions, Proc. Imp. Acad.
Tokyo 20 (1944), 706-714.
10. B. J. Maitland, A note on functions regular and bounded in the unit circle and small at a set of
points near the circumference of the circle, Proc. Cambridge Philos. Soc. 35 ( 1939), 382-388.

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

 HARMONIC MEASURE AND RADIAL PROJECTION 95

11. D. Marshall and J. Morrow, Unzipping plane domains (numerical computations of conformai
maps), preprint.
12. H. Milloux, Le théorème de M. Picard, suites de fonctions holomorphes, fonctions méromorphes
et fonctions entières, J. Math. 3 (1924), 345-401.
13. _, Sur le théorème de Picard, Bull. Soc. Math. France 53 (1925), 181-207.
14. R. Nevanlinna, Über eine Minimumaufgabe in der Theorie der konformen Abbildung, Göttinger
Nachr. 1.37(1933), 103-115.
15. L. N. Trefethen, Conformai map of a rectangle, informal notes, 1983.
16. _, Analysis and design of polygonal resistors by conformai mapping, Z. Angew. Math. Phys.
35(1984), 692-704.

Department of Mathematics, University of Washington, Seattle, Washington 98195

Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37916

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